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Multivariate power series multiplication
 IN ISSAC’05
, 2005
"... We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estima ..."
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We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.
CommunicationOptimal Parallel Recursive Rectangular Matrix Multiplication
, 2012
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Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication
, 2012
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Multihomogeneous polynomial decomposition using moment matrices
 International Symposium on Symbolic and Algebraic Computation
, 2011
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 7 (4 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Efficient Algorithms for Path Problems in Weighted Graphs
, 2008
"... Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shorte ..."
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Cited by 5 (0 self)
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Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the allpairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat exten ..."
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Cited by 3 (2 self)
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The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
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"... Using KML files as encoding standard to explore locations, access and ..."
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Using KML files as encoding standard to explore locations, access and
TENSORSPARSITY OF SOLUTIONS TO HIGHDIMENSIONAL ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
"... Abstract. A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to highdimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the po ..."
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Abstract. A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to highdimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class Σn of functions, which can be written as a sum of rankone tensors using a total of at most n parameters and then uses this notion of sparsity to prove a regularity theorem for certain highdimensional elliptic PDEs. It is shown, among other results, that whenever the righthand side f of the elliptic PDE can be approximated with a certain rate O(n−r) in the norm of H−1 by elements of Σn, then the solution u can be approximated in H1 from Σn to accuracy O(n−r ′ ) for any r ′ ∈ (0, r). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second “basisfree ” model of tensor sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model highdimensional elliptic PDEs with tensorsparse data. 1.